Given a locally finitely presentable category $\mathcal{C}$ it is well-known that every functor category $[\mathcal{A},\mathcal{C}]$ (where $\mathcal{A}$ is a small category) is also locally finitely presentable. Is there a concrete description of the finitely presentable objects in such categories?

If this is not possible in general, what about special cases like $\mathcal{A}=\mathcal{Set}_{fin}^{op}$?